Chapter One: Rates and Variations - Mathematics Form Two

Chapter One: Rates and Variations – Mathematics Form Two

Introduction

Understanding rates and variations is essential for grasping how quantities change in relation to each other. Some variables are interdependent, meaning a change in one affects the others. Many real-life situations involve such relationships.

In this chapter, you will describe the concepts of rates and variations, explain the types of variations, and solve problems on rates and variations. The competence developed will enable you to solve real-life challenges such as finding shopping deals, assessing fuel efficiency, or predicting data trends in fields like economics, science, engineering, and many other applications.

Rates

Rates show how one quantity is related to another quantity either increasing or decreasing its quantity. For example, speed of moving object is the ratio between the distance covered and time taken.

Activity 1: Exploring rates in real life

Engage in Activity 1.1 to explore the concept of rates in real life.

Example 1

If a car travels 150 kilometres in 3 hours. What is its speed?

Solution

Given distance = 150 km and time = 3 hrs.

But, speed is the rate of change of distance with respect to time. That is,

speed = distance/time

Thus, speed = 150 km / 3 hrs = 50 km per hour

Therefore, the rate of travel of the car is 50 km in every hour.

Example 2

If a water tank fills up with 200 litres of water in 5 minutes, what is the rate of flow of water?

Solution

Given the volume of water is 200 L and the time the tank takes to be filled is 5 minutes.

Rate of flow = Volume / Time

Rate of flow = 200 L / 5 min = 40 L/min

Therefore, water enters the tank at the rate of 40 litres per minute.

Example 3

A student had two plant seedlings. She measured the rate at which the seedlings were growing. Seedling A grew 5 cm in 10 days and seedling B grew 8 cm in 12 days. Which seedling was growing more quickly?

Solution

The rates of growth of the two seedlings is computed as follows:

Rate of growth of seedling A = 5 cm / 10 days = 0.5 cm per day

Rate of growth of seedling B = 8 cm / 12 days = 0.67 cm per day

The growth rate of seedling B is higher than that of seedling A.

Therefore, seedling B was growing more quickly than seedling A.

Example 4

Two pipes, A and B are used to fill a water tank. Pipe A can fill the tank in 6 hours, while pipe B can fill the same tank in 4 hours. If both pipes are opened at the same time, how long will it take to fill the tank?

Solution

Pipe A fills the tank in 6 hours, so it fills 1/6 of the tank in 1 hour.

Pipe B fills the tank in 4 hours, so it fills 1/4 of the tank in 1 hour.

Combined rate = Rate of A + Rate of B

Combined rate = 1/6 + 1/4 = (2 + 3)/12 = 5/12

Therefore, both pipes together fill 5/12 of the tank in 1 hour.

If 5/12 of the tank is filled in 1 hour, then the time taken to fill the whole tank will be:

Time = Whole tank / Combined rate = 1 / (5/12) = 12/5 = 2.4 hours

Therefore, it will take 2.4 hours to fill the tank when both pipes are opened simultaneously.

Exercise 1

What rate in metres per second is equivalent to a speed of 45 kilometres per hour?
A car covers a distance of 200 kilometres in 60 minutes. What is the speed of the car in metres per second?
A water tap takes 10 minutes to fill a 500 litres tank. Find the rate of flow of water in litres per second.
If premium petrol costs Tsh 2,500 per litre, how many litres can be purchased for Tshs 10,000?
Find the rate in kilometres per hour if a racing car travels 115½ kilometres in 35 minutes.
Mwajuma walks 6 kilometres in 1 hour and John walks the same distance in 2 hours. What are their walking speeds? Who walks faster, and by how much?
A garden hose fills a 20 litre bucket in 4 minutes. What is the rate of flow in litres per minute?
A car travels 400 km using 25 litres of fuel. What is the car’s fuel efficiency in kilometres per litre?
Mr. Magoda’s salary increased from Tsh 800,000 to Tsh 1,000,000 in a year. What is the rate of change of his salary?
The temperature in Mrs. Kidunula’s room increased from 15°C at 8 am to 25°C at 2 pm. What was the average rate of temperature change per hour?
A car accelerates from 0 to 60 km/h in 10 seconds. What is the rate of change of its speed in km/h per second?
Mr. Kilenzi’s heartbeat increased from 60 beats per minute to 120 beats per minute during exercise over 3 minutes. What was the average rate of change in his heartbeats?
A family uses 80 GB of internet data in 30 days. What is the average daily rate of data usage in GB per day?
Pipe X can fill a tank in 8 hours, while Pipe Y can fill it in 5 hours. How long will it take to fill the tank if both pipes are opened simultaneously?
Pump A can fill a pool in 5 hours, Pump B in 4 hours, and Pump C in 10 hours. How long will it take to fill the pool if all three pumps are operated at the same time?

Exchange Rates

In any country, people expect to do transactions in the currency of their own country. When money from country A is to be used in country B, it is necessary to exchange the currency of country A to the currency of country B. Various currencies in the world are linked together by exchange rates. This enables smooth transfer of money and payments to take place between countries.

Activity 2: Performing currency exchange rates

Choose a suitable computer Currency Conversion Application.
Analyse current exchange rates between the Tanzanian currency and other currencies of your choice using the computer application.
Compare the app’s rates with market rates and evaluate its accuracy for financial decisions, especially for travel.

Exercise 2

Use any computer Currency conversion application to answer questions 1 ~ 10.

How much is Tsh 20,600 worth in Indian Rupees?
Convert 1 New Zealand Dollar into Saudi Arabia Riyal.
How much is Tsh 500,000 worth in Euros?
How many Yen are equivalent to Tsh 1?
How many Tanzanian shillings can a visitor from Kenya get for exchanging 930 Kenyan shillings?
Find the amount in Tanzanian shillings of each of the following:
1. 30,000 Euros
2. 4,200 Pula
3. 640 Rands
4. 12,000 Riyal
Exchange Tsh 300,000 into the following currencies:
1. Mozambican meticals
2. Malawian kwachas
3. Swiss francs
4. Indian rupees
How much is Tsh 6,000,000 worth in Pounds Sterling?
Mwanaisha bought story books for 200 AUD (Australian Dollars). How much did she spend in Tanzanian shillings?
Mr. Utaligolo wants to exchange 1,000 USD for EUR. If the exchange fee is 2%, how much will he receive after charging the fee?
Mrs. Uwemba is shopping in a country where 1 USD = 205 Local Currency (LC). If she spent 205,000 LC, how much did she spend in USD?
Ayota Stationery wants to buy items from an online store, which cost 2500 NOK (Norwegian Kroner). If the exchange rate is 1 NOK = 9 Local Currency, how much does the stationery pay in Local Currency?

II. Variations

Variation is a relationship where a change in one quantity leads to a proportional change in the other. It allows for the exploration of connections between two or more quantities. The four basic types of variations are direct, inverse, joint, and combined.

Direct Variations

Direct variation is a relationship between two variables where one variable is a constant multiple of the other. In other words, if one variable increases or decreases, the other variable changes proportionally in the same direction.

This type of variation is useful for understanding how changes in one variable affect another in a directly proportional manner.

Activity 3: Exploring direct variation in real life

Learn about direct variation from books or from the internet.
Find real-life examples that illustrate direct variations.
Demonstrate mathematically how variables in the real-life scenarios relate and use the relationship to solve related problems.

If y is directly proportional to x, it can be written as y ∝ x, where ∝ is a symbol of proportionality. The corresponding mathematical equation connecting x and y is formed by introducing a proportionality constant k, and replaces ∝ with an equal sign to get, y = kx.

For instance, if y varies directly as the square of x, then y ∝ x² and the corresponding equation is y = kx², where k is the constant of proportionality.

For any two pairs of quantities x and y, say (x₁, y₁) and (x₂, y₂), two equations y₁ = kx₁² and y₂ = kx₂² are obtained. This implies that k = y₁/x₁² = y₂/x₂². So, it is said that x and y vary directly if the ratios of the values of y to the values of x are proportional.

If x and y are any two quantities that are in direct variation, then y = kx. The nature of the equation y = kx is a straight line passing through the origin, where k represents the gradient (slope) of the line.

Example 5

If y varies directly as x and x = 15 when y = 4, find the value of y when x = 12.

Solution

Given that y ∝ x. This implies that y = kx, where k is the constant of proportionality.

Making k the subject of the equation gives k = y/x

But, for any two pairs of quantities, k = y₁/x₁ and k = y₂/x₂.

Thus, y₁/x₁ = y₂/x₂

Given x₁ = 15, x₂ = 12, and y₁ = 4, the value of y₂ is obtained as follows:

y₂ = (y₁ × x₂)/x₁ = (4 × 12)/15 = 48/15 = 16/5

Therefore, the value of y is 16/5 when x = 12.

Example 6

If x varies directly as the square of y and x = 4 when y = 2, find the value of x when y = 8.

Solution

Let x₁ = 4, y₁ = 2 and y₂ = 8. The variation equation is given by:

x₁/x₂ = y₁²/y₂²

Substituting the values of x₁, y₁, and y₂ gives:

4/x₂ = 2²/8² = 4/64

Cross multiplication gives:

4 × x₂ = 4 × 64

x₂ = (4 × 64)/4 = 64

Therefore, the value of x is 64 when y is 8.

Example 7

A car travels 60 kilometres using 5 litres of diesel. How many litres of diesel are needed to travel 150 kilometres?

Solution

Let x denote the number of litres of diesel and y denote the number of kilometres.

The equation for the variation becomes x₁/x₂ = y₁/y₂

Given x₁ = 5 litres, y₁ = 60 km, and y₂ = 150 km, the value of x₂ is given by:

x₂ = (x₁ × y₂)/y₁ = (5 litres × 150 km)/60 km = 750/60 = 12.5 litres

Therefore, 12.5 litres of diesel are needed to travel 150 kilometres.

Exercise 3

If x varies directly as y and x = 16 when y = 10, find the value of y when x = 20.
The surface area of a circular object varies directly as the square of its radius. If its surface area is 78.5 cm² when the radius is 5 cm, find the surface area of the circular object when the radius is 7 cm.
If x varies directly as y and x = 30 when y = 40, find the value of x when y = 16.
A mason can build 100 metres of fence in 20 hours. How long will it take 5 masons with the same ability to build 875 metres of fence?
If x varies directly as 2y + 7 and x = 5, when y = 4, find the value of y when x = 6.
If 8 men can assemble 16 machines in 12 days, how long will it take 15 men of the same ability to assemble 100 machines?
If y varies directly as the square root of x, and y = 12 when x = 4, find the value of y when x = 9.
If y varies directly as x and y = 8 when x = 3, find the value of y when x = 18.
Two variables x and y that vary directly have corresponding values as shown in the following table.

x 3 5 6 8

y 17 34 58
1. Find the rule connecting x and y.
2. Fill in the missing values.
3. Draw a graph which shows that y ∝ x for k = 1.
Study the following table and answer the questions that follow.

Hours worked 2 3 4 5

Earning (Tsh) 1,150 1,725 2,300 2,875
1. Do earnings vary directly as the number of hours worked?
2. If yes, calculate the constant of proportionality and find the equation that describes the relationship.
The volume of a sphere varies as the cube of its radius. Three solid spheres of diameters 3/2 m, 2 m, and 5/2 m are melted and combined to form a new solid sphere. Find the diameter of the new sphere.
When observing two buildings simultaneously, the length of each building’s shadow varies directly with its height. If a 5 floor building has a shadow of length 20 m, how many floors of a building would form a shadow of length 32 m?
The resistance of a wire varies as the square of the diameter of its cross-section. Find the percentage change in the resistance when the diameter is (a) doubled (b) reduced by 20%.
Two variables A and x are related by the formula A = axⁿ. The following set of data was generated based on this formula.

x 1 2 3 4

A 0.5 2 4.5 8
1. Find the values of a and n.
2. Find the value of A when x = 5.
A precious stone worth Tsh 15,600,000 is accidentally dropped and broken into three pieces. The weights of the pieces are in the proportions of 2:3:5, respectively. If the value of the precious stone varies directly as the cube of its weight, calculate the value of the remaining stone in percentage.

x	3	5	6	8
y	17		34	58

Hours worked	2	3	4	5
Earning (Tsh)	1,150	1,725	2,300	2,875

x	1	2	3	4
A	0.5	2	4.5	8

Inverse Variations

A relationship between two or more variables is said to be an inverse variation if the value of one variable increases while the other value decreases, or vice versa.

Activity 4: Identifying inverse variations in daily life

Identify three real-life activities where increasing one variable decreases the other variable.
Perform one of the activities to experience how changes of the variables are related in the activity.
Find out through reading books and browsing the internet how to express these inverse relationships mathematically.
Use the mathematical expression to explain how changes in one variable affect the other, and present your findings.

Quantities with an inverse variation relationship are said to be inversely proportional to each other. In this case, the quantities vary inversely or in inverse proportion. Inverse proportion is sometimes referred to as indirect proportion.

For example, the number of men employed to cultivate a farm and time taken to complete the work are inversely related. Likewise, the time to travel to a certain place and the speed are inversely related.

Generally, if y has an inverse relationship with x, then y is proportional to the reciprocal of x. This relationship is denoted by:

y ∝ 1/x

The equation relating y and x is formed by introducing a constant of proportionality, k and replacing the symbol of proportionality with an equal sign (=). That is, the inverse variation between y and x becomes:

y = k(1/x)

If y varies inversely as the square of x, the equation connecting x and y is y = k/x² or k = yx².

Example 1.8

If x varies inversely as y and x = 2 when y = 3, find the value of y when x = 18.

Solution

The statement x ∝ 1/y implies that x = k/y, where k is the constant of proportionality.

Making k the subject of the equation gives, xy = k, which implies that x₁y₁ = x₂y₂.

When x₁ = 2, y₁ = 3, and x₂ = 18, the value of y₂ is:

y₂ = (x₁y₁)/x₂ = (2 × 3)/18 = 6/18 = 1/3

Therefore, the value of y is 1/3 when the value of x is 18.

Example 9

If it takes 12 days for 10 men to assemble a machine, how long does it take 15 men with the same ability to assemble the same machine?

Solution

Let m be the number of men and d be the number of days. It is obvious that 15 men will take less time to assemble the machine than 10 men.

Thus, m and d vary inversely, that is, m ∝ 1/d, which implies that m = k/d.

Thus, k = md.

For the two pairs of quantities (m₁, d₁) and (m₂, d₂), it follows that, m₁d₁ = m₂d₂.

Thus, d₂ = (m₁d₁)/m₂

Given d₁ = 12 days, m₁ = 10 men and m₂ = 15 men, the value of d₂ is:

d₂ = (10 men × 12 days)/15 men = 120/15 = 8 days

Therefore, it takes 8 days for 15 men to assemble the same machine.

Example 10

The intensity of light is inversely proportional to the square of the distance D from the light source. Calculate the percentage change in intensity under the following conditions:

The distance is halved.
The distance is increased by 30%.

Solution

I ∝ 1/D²

I = k/D²

(a) If the distance D is halved, the new distance is D₁ = D/2.

The new intensity I₁ = k/(D/2)² = k/(D²/4) = 4k/D²

Percentage change = [(I₁ – I)/I] × 100% = [(4k/D² – k/D²)/(k/D²)] × 100% = 300%

Therefore, the intensity increases by 300%.

(b) If the distance is increased by 30%, the new distance is D₁ = 1.3D.

The new intensity I₁ = k/(1.3D)² = k/(1.69D²) = (1/1.69)(k/D²)

Percentage change = [(I₁ – I)/I] × 100% = [((1/1.69)(k/D²) – k/D²)/(k/D²)] × 100% = (1/1.69 – 1) × 100% ≈ -40.83%

Therefore, the intensity decreases by approximately 40.83%.

Joint and Combined Variations

In some activities, one variable can depend on several other variables to operate effectively. Such relationships are described by joint and combined variations.

Activity 5: Discovering joint and combined variations in daily life

Explore the concepts of joint and combined variations using books and online resources.
Find and describe real-life examples of such variations using daily practices such as formulas.
Share your final observations and discuss the examples you discovered.

Joint Variation

Consider a formula for finding the area of a triangle. It is given by Area = 1/2 × base × height. In this equation, the area of a triangle varies directly as the product of its base and height. Variable relationships of this nature are known as joint variations. This specific variation can be generally expressed as Area ∝ base × height.

A joint variation occurs when a variable is directly or inversely proportional to the product of two or more variables. Mathematically, if a variable z varies jointly as x and y, the general formula for joint variation can be written as:

z ∝ xy ⇒ z = kxy

Example 11

Suppose y varies directly as x and z. Given x = 4, z = 2, and y = 24, find:

The variation equation connecting x, y, and z.
The value of y when x = 5 and z = 6.

Solution

(a) Since y ∝ x and y ∝ z, it follows that y ∝ xz, y = kxz.

The variation equation is k = y/(xz), where k is a constant of proportionality.

Given x = 4, z = 2, and y = 24, it follows that k = 24/(4 × 2) = 24/8 = 3

Therefore, the variation equation is y = 3xz.

(b) With y = 3xz, if x = 5 and z = 6, then y = 3 × 5 × 6 = 90

Therefore, the value of y is 90.

Example 12

A bakery has a project to bake 240 cakes. With 4 bakers working for 5 days, they can complete the task. If the number of cakes is increased to 300 and the bakery decides to use 6 bakers, how many days will it take to bake all 300 cakes?

Solution

Let c be the number of cakes, b be the number of bakers, and d be the number of days.

More cakes can be produced if there are more bakers and more working days, assuming other factors remain constant.

It implies that these variables vary jointly and are directly related. Thus, c ∝ bd.

c = kbd.

Given c = 240, b = 4, and d = 5, it implies that 240 = 4 × 5 × k = 20k

k = 240/20 = 12

From c = kbd, c = 12bd

Given, c = 300, b = 6, 300 = 12 × 6 × d = 72d

d = 300/72 = 4.2

Therefore, 4.2 days will be needed to bake 300 cakes or it requires 4 days, 4 hours, and 48 minutes to bake 300 cakes.

Combined Variation

Combined variation occurs when a variable depends on two or more other variables, with some relationships being direct and others inverse.

Example 16

The cost of a certain material varies directly with the quantity purchased and inversely with the number of suppliers. If the cost is Tshs 120,000 when the quantity is 100 units and there are 4 suppliers, find the constant of variation and the cost when the quantity becomes 150 units and there are only 3 suppliers.

Solution

Let C be the cost of materials, q be the quantity purchased, and S be the number of suppliers.

Thus, C ∝ q and C ∝ 1/S. The combined variation is:

C ∝ q/S ⇒ C = k(q/S)

Given C = Tshs 120,000, q = 100, S = 4, then

k = (C × S)/q = (120,000 × 4)/100 = 480,000/100 = 4,800

Now, if S = 3 and q = 150, it follows that

C = (4,800 × 150)/3 = 720,000/3 = 240,000

Therefore, the cost of materials will be Tshs 240,000.

Exercise 5

If y varies directly as the square of x and inversely as z, find the percentage change in y when x is increased by 10% and z is decreased by 20%.
Suppose P varies directly as V and inversely as the square root of R. If P = 180 when R = 25 and V = 9, find the value of P when V = 6 and R = 36.
The height h of a cone varies directly as its volume V and inversely as the square of its radius r. Write a formula for the height of the cone.
If y² varies directly as x-1 and inversely as x+d and x = 2, d = 4 when y = 1, find the value of x when y = 2 and d = 1.
If two typists in a typing pool can type 210 pages in 3 days, how many typists working at the same speed will be needed to type 700 pages in 2 days?
Suppose x varies directly as y² and inversely as p. If x = 2, when y = 3 and p = 1, find the value of y when x = 4 and p = 5.
If V varies directly as the square of x and inversely as y, and if V = 18 when x = 3 and y = 4, find the value of V when x = 5 and y = 2.
Use a mathematical software to draw the following curves. Assume that the constant of proportionality is 1.
1. y ∝ 1/x
2. y ∝ 1/x²
3. y ∝ 1/√x
The following table shows the values of y for some selected values of x. The variables x and y are connected by the relation, ‘y varies inversely as x’. Calculate the missing values of y.

x 5 10 15

y a 3 b 1.5
Express each of the following relations as an equation using k as a constant of proportionality.
1. c varies directly as p and q, and inversely as s.
2. d varies jointly as t and r².
3. d varies directly as y and the square root of z.
The heating cost H for a house varies directly with its size S in square metres and inversely with the efficiency rating E of the heating system. If H = 50,000 shillings for a house of 200 square metres with an efficiency rating of 4, find k and the heating cost for a house of 250 square metres with an efficiency rating of 5.
The shipping cost C varies directly with the mass W of the package and inversely with the number of packages n. If the cost is TShs 7,000,000 for a package weighing 20 kg, and 6 packages, find k and the cost of a package weighing 30 kg with 5 packages.

x	5	10	15
y	a	3	b	1.5

Chapter Summary

A rate gives the change of one quantity with respect to another quantity.
Exchange rate is the conversion rate between different currencies.
Variation is the relationship in which the change in one quantity results in a proportional change in the other.
If y = kx, then y varies directly with x, or y is directly proportional to x. The constant k is called a constant of proportionality.
If y varies as 1/x, then y is inversely proportional to x.
If a quantity varies as the product of two or more quantities, then it varies jointly with other quantities.
If both direct variation and inverse variation occur at the same time, then it is called a combined variation.

Revision Exercise

If y = kx and y = 8 when x = 7, find the value of k and the value of y when x = 40.
If y is directly proportional to x and y = 10 when x = 4, find the value of y when x = 15 and the value of x when y = 8.4.
If y ∝ x and y = 16.5 when x = 3.5, find the equation connecting x and y. Hence, find the value of x when y = 21.
If y is proportional to x² and if x = 15 when y = 200, find the equation connecting x and y. Find the value of y when x = 8.5.
If y ∝ √x and y = 3.5 when x = 4, express y in terms of x. What is the value of y when x = 25?
If y ∝ 1/x, find the values of a, b, and c in the following table.

x a 1.2 8

y 6 b 1.5 0.8
Given that y varies directly as x and inversely as z. If y = 10 when x = 8 and z = 5, find the equation connecting x, y, and z. Find the value of y when x = 6 and z = 2.5.
If y varies jointly as x and z², and if y = 13⅓ when x = 2.5 and z = 4/3, find the equation connecting the three variables. Find the value of x when z = 3/2 and y = 54.
Suppose y varies directly as x² and inversely as √z. If x = 8, y = 16, and z = 25, find the value of y when x = 5 and z = 9.
Determine whether the data in the following tables have an inverse variation relationship. If yes, find the missing values.
1. x 7 9 12 6
  
  y 10 12 15 12
2. x 12 6 21 3
  
  y 4 2 7 3
3. x -15 -8 10
  
  y -8 -15 10
A dairy farm dispenses milk into a container at a rate of 45 litres per minute.
1. How much milk is dispensed in 2 hours?
2. How long will it take to empty a 2000 litre tank?
A water pump operates at a flow rate of 300 litres per minute.
1. Calculate the total volume of water pumped in 2 hours.
2. Calculate the time required to fill a 30,000-litre tank.
A truck travels 800 km and uses 90 litres of diesel.
1. Calculate the fuel consumption rate in litres per 100 km.
2. How much diesel will the truck use to travel 600 km at the same rate?
3. If the truck travels 1,300 km, how much diesel will it consume?
State whether distance and speed vary directly or inversely. Give reason.
If x and y vary inversely, use the given pair of values to find an equation which in each case relate the variables:
1. x = 6, y = 4
2. x = 8, y = 12
3. x = 10, y = 1/3
A group of 10 volunteers can pack 400 boxes of food in 5 hours. If the organization decides to pack 600 boxes and increases the number of volunteers to 12, how many hours will it take to pack all 600 boxes?
A publishing company needs to print 2,000 copies of a book. With 4 printers, the task is completed in 10 days. If the company decides to print 3,000 copies and uses 6 printers, how many days will it take to complete the printing?
A logistics team of 8 members can package and ship 400 orders in 5 days. If the number of team members is increased to 12 and the number of orders to be shipped is raised to 600, calculate the number of days needed to complete the work.
The intensity I of light, in lux, on a surface varies directly with the power P in watts of light and inversely with the square of the distance d from the light source. A photographer is setting up lighting for a photoshoot. If the light intensity is 400 lux when a light source of 66 watts is placed 2 m away, what will be the intensity when a light source of 100 watts is placed 3 m away?
Machine A can produce 800 parts per hour, while Machine B can produce 1,000 parts per hour. There is an order of 24,000 parts.
1. How long will the machines take working together to process the order?
2. Machine B was on preventive maintenance, so Machine A had to work alone for the first 5 hours before Machine B joins in. How many more hours will they have to work together to finish the job?
Pipe A can fill a pond in 8 hours, while Pipe B can fill it in 3 hours. However, Pipe B has a defect that causes it to lose water at the rate of 1 litre every 20 minutes while filling. If the pond has a capacity of 200 litres and both pipes are opened simultaneously, how long will it take to fill the pond?

x	a	1.2	8
y	6	b	1.5	0.8

x	-15	-8	10
y	-8	-15	10